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March 16th, 2012, 08:46 PM   #1
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Price Elasticity, Separation of Variables

So I naively took price elasticity

http://en.wikipedia.org/wiki/Price_elasticity_of_demand
http://en.wikipedia.org/wiki/Elasticity_%28economics%29

Ed = ((Qd-Qo)/Qd)/((Pd-Po)/Pd))

In the differential form:

Ed = (P/Q)*dQ/dP

Separated variables:

(Ed/P)dP = (1/Q)dQ

Integrated the left from Po to P and the right from Qo to Q and got the incredibly borrowing result:

exp(Ed)(P/Po)=Q/Qo

Which essentially says that the only functions that can satisfy constant price elasticity are a non negative slopping line though the origin.

Now, this result is disappointing because this does not look like a typical demand curve at all, yet price elasticity is used to consider effects on total revenue due to changes in quantities:

http://en.wikipedia.org/wiki/Total_revenue_test

I was expecting a function that would be able to interpolate a demand function well so I could approximate a small a small region of the curve with constant elasticity. I could use the non differential form but I was expecting them to agree over a small region and am wondering what I might have done wrong.

Reviewing separation of variables:
http://en.wikipedia.org/wiki/Separation_of_variables

I see I forgot an absolute value sign. This will also gives me negative slope solutions but I'm still restricted to lines that go though the origin.

: Pauses and scratches head :
John Creighton is offline  
 
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